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% On the algebra of coupled SO(3) tensors
% K. Srinivasa Rao, J. Van der Jeugt and G. Vanden Berghe
% J. Math. Phys. 33 (1992), 15-18.
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\begin{center}
{\LARGE On the algebra of coupled SO(3) tensors}\\[2cm]
K. Srinivasa Rao$^{a)}$\,,\ J. Van der Jeugt$^{b)}$ and G. Vanden Berghe
\\[.5cm]
{\em Laboratorium voor Numerieke Wiskunde en Informatica,\\
Rijksuniversiteit Gent, Krijgslaan 281--S9, B-9000 Gent, Belgium}\\
\end{center}


\vspace{3cm}
\noindent \underline{Abstract}

\begin{minipage}{14cm}
The commutation relation for coupled SO(3) tensor operators is constructed.
This gives rise to a Lie algebra for which $9j$ symbols appear in the structure
constants. In an example it is shown how an exceptional Lie subalgebra can be
realised. The closure requirement for the exceptional Lie algebra does not seem
to be related to non--trivial $9j$ symbols but rather to vanishings of relations
involving $6j$ symbols.
\end{minipage}

\vfill
\noindent-----------------------------------\\
$^{a)}$ {\footnotesize Permanent address~:
The Institute of Mathematical Sciences, Madras-600\,113, India}\\
$^{b)}$ {\footnotesize Research Associate of N.F.W.O.
(National Fund for Scientific Research of Belgium)}
\newpage

Let $T^k_q(j_1 j_2)$ $(q=-k,\,-k+1,\,\ldots,\,+k)$ denote a set of SO(3) tensor
operators of rank $k$ mapping the $(2j_2+1)$-dimensional vector space with angular
momentum basis $|j_2\,,\,m_2\rangle $ into the $(2j_1+1)$-dimensional vector space with
basis vectors $|j_1\,,\,m_1\rangle $. Such tensor operators were originally introduced
by Elliott $^1$ in his description of collective motion in the nuclear shell
model. These tensor operators are completely defined by means of their reduced
matrix element
\be
\langle \, j_b\parallel T^k(j_1\,j_2)\parallel j_a\,\rangle
= [k]^{1/2} \delta_{j_1j_b}\delta_{j_2j_a}
\ee
where $[k]$ stands for $(2k+1)$.
Judd $^2$ calculated the commutation relations between the SO(3) tensor operators
$T^k_q(jj)$ (i.e. the case $j_1=j_2=j$), which was later extended to the general
case by Vanden Berghe and De Meyer $^3$ to the following expression~:
\bea \lefteqn{
\left[ T^k_q(j_1j_2),T^{k'}_{q'}(j_3j_4)\right]
= \sum_{k'',q''}[k\,k'\,k'']^{1/2}
  \left( \begin{array}{ccc} k & k' & k''\\
                            q & q' & -q''
         \end{array} \right)
  (-1)^{2j_4 + j_3-j_2 -q''} }\nn\\
&& \times \left( \delta_{j_2j_3}(-1)^{k+k'+k''+j_1+j_2+j_3+j_4}
  \left\{ \begin{array}{ccc} k & k' & k''\\
                            j_4 & j_1 & j_2
          \end{array} \right\} T^{k''}_{q''}(j_1j_4)\right.\nn\\
&&\qquad\left. - \delta_{j_1j_4} \left\{ \begin{array}{ccc} k & k' & k''\\
                                             j_3 & j_2 & j_1
                           \end{array} \right\}
   T^{k''}_{q''}(j_3j_2) \right)\,.
\eea
In (2), the standard notation for SO(3) $3j$ and $6j$ symbols $^4$ is used.
The commutation relation (2) has been used to provide realisations of certain
Lie algebras. By identifying some exceptional Lie subalgebras, a relation
between so-called non-trivial zeros $^5$ of $6j$ symbols appearing in (2) and
embeddings of exceptional Lie algebras has been established $^{6,7,8}$.

It is the purpose of the present note to extend the commutation relation (2)
to the case of coupled tensor operators. The $9j$ symbol seems to play a
key role in this situation. The Lie algebraic structure of coupled tensor
operators is discussed, and some possible applications are commented on.
%
%3
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Coupled SO(3) tensor operators are of the form
\be
\Bigl( T^{k_1}(j_1j_2) \otimes T^{k'_1}(j'_1j'_2)\Bigr)^k_q
= \sum_{q_1,q'_1} \langle k_1\;q_1\;k'_1\;q'_1|k\;q\rangle
T^{k_1}_{q_1}(j_1j_2)\otimes T^{k'_1}_{q'_1}(j'_1 j'_2)\,,
\ee
where $\langle k_1\;q_1\;k'_1\;q'_1|k\;q\rangle $ is an SO(3) coupling coefficient $^4$,
and the SO(3) tensor operators act on independent angular momentum vectors~:
\bea \lefteqn{
T^{k_1}_{q_1}(j_1j_2) \otimes T^{k'_1}_{q'_1}(j'_1 j'_2)
(|j_a\,m_a\rangle  \otimes |j'_a\, m'_a \rangle  )= }\nn\\
&& T^{k_1}_{q_1}(j_1j_2)|j_a\,m_a \rangle  \otimes
   T^{k'_1}_{q'_1}(j'_1j'_2)|j'_a\,m'_a \rangle \,.
\eea
The action of a tensor operator on an angular momentum state follows from (1)
and the Wigner-Eckart theorem~:
\be
T^k_q(j_1j_2)|j_a\,m_a\rangle  =
\sum_{j_bm_b} (-1)^{j_b-m_b}
\left( \begin{array}{ccc} j_b & k & j_a\\
                         -m_b & q & m_a \end{array}\right)
[k]^{1/2}\delta_{j_1j_b}\, \delta_{j_2 j_a}|j_bm_b\rangle \,.
\ee
In order to find an expression for the commutator of two coupled SO(3)
tensor operators, one calculates the action of their product on a vector
$|j_a\,m_a\rangle \otimes |j'_a\,m'_a\rangle $\,, i.e.
\be
\Bigl( T^{k_1}(j_1j_2) \otimes T^{k'_1}(j'_1j'_2) \Bigr)^k_q
\Bigl( T^{k_2}(j_3j_4) \otimes T^{k'_2}(j'_3j'_4) \Bigr)^{k'}_{q'}
(|j_a\,m_a\rangle  \otimes |j'_a\,m'_a\rangle )\,.
\ee
To determine (6), the couplings $(k_1\,k'_1\,k)$ and $(k_2\,k'_2\,k')$ are
first decoupled, giving rise to two $3j$ symbols
$\left( \begin{array}{ccc}k_1 & k'_1 & k \\ q_1 & q'_1 & -q \end{array} \right)$
and
$\left( \begin{array}{ccc}k_2 & k'_2 & k' \\ q_2 & q'_2 & -q' \end{array} \right)$.
Then, the following expression for the product of two tensor operators,
deduced from (5), is used~:
\bea \lefteqn{
T^{k_1}_{q_1}(j_1j_2)T^{k_2}_{q_2}(j_3j_4)|j_a\,m_a\rangle  =
\sum_{K,Q} \delta_{j_2j_3}(-1)^{2j_2+j_4-j_1-Q+k_1+k_2+K} }\nn\\
&& [k_1\;k_2\;K]^{1/2}
\left( \begin{array}{ccc}
       k_1 & k_2 & K\\
       q_1 & q_2 & -Q \end{array} \right)
\left\{ \begin{array}{ccc}
       k_1 & k_2 & K\\
       j_4 & j_1 & j_2 \end{array} \right\}
T^K_Q(j_1j_4)|j_a\,m_a\rangle \,.
\eea
%
%4
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Applying (7) both for the $|j_a\,m_a\rangle $ state and the $|j'_a\,m'_a\rangle $ state
yields the two $3j$ symbols
$\left( \begin{array}{ccc} k_1 & k_2 & K\\
                           q_1 & q_2 & -Q \end{array} \right)$
and
$\left( \begin{array}{ccc} k'_1 & k'_2 & K'\\
                           q'_1 & q'_2 & -Q' \end{array} \right)$.
Finally, recoupling $T^K_Q(j_1j_4)\otimes T^{K'}_{Q'}(j'_1\,j'_4)$ to a coupled
tensor of rank $k''$ gives rise to the $3j$ symbol
$\left( \begin{array}{ccc} K & K' & k''\\
                          -Q & -Q'& q'' \end{array} \right)$.
With these five $3j$ symbols, the following summation can be performed $^9$~:
\bea
&&
{\small
\addtolength{\arraycolsep}{-.5\arraycolsep}
\sum_{\scriptstyle q_1 , q'_1, q_2 , q'_2 , Q , Q'}
\left( \begin{array}{ccc} k_1 & k'_1 & k\\
                          q_1 & q'_1 & -q \end{array} \right)
\left( \begin{array}{ccc} k_2 & k'_2 & k'\\
                          q_2 & q'_2 & -q' \end{array} \right)
\left( \begin{array}{ccc} k_1 & k_2 & K\\
                          q_1 & q_2 & -Q \end{array} \right)
\left( \begin{array}{ccc} k'_1 & k'_2 & K'\\
                          q'_1 & q'_2 & -Q' \end{array} \right)
\left( \begin{array}{ccc}  K & K'  & k''\\
                          -Q & -Q' & q'' \end{array} \right) } \nn\\
&& \qquad\qquad\qquad
 = \left( \begin{array}{ccc} k & k' & k''\\
                          -q & -q' & q'' \end{array} \right)
\left\{ \begin{array}{ccc} k_1 & k'_1 & k\\
                           k_2 & k'_2 & k'\\
                           K & K' & k'' \end{array} \right\}\,.
\eea
Hence, keeping track of all the phase factors, one finds~:
\bea \lefteqn{
\Bigl( T^{k_1}(j_1j_2) \otimes T^{k'_1}(j'_1j'_2)\Bigr)^k_q
\Bigl( T^{k_2}(j_3j_4) \otimes T^{k'_2}(j'_3j'_4)\Bigr)^{k'}_{q'}
|j_a\,m_a\rangle  \otimes |j'_a\,m'_a\rangle  = }\nn\\
&& \sum_{K,K',k'',q''}
  (-1)^{2j_2-j_1+j_4+2j'_2-j'_1+j'_4+K+K'+k''+q''}
  [k_1\,k_2\,K\,k'_1\,k'_2\,K'\,k\,k'\,k'']^{1/2}\delta_{j_2j_3}\delta_{j'_2j'_3}\nn\\
&& \times
\left( \begin{array}{ccc} k & k' & k''\\
                          -q & -q' & q'' \end{array} \right)
\left\{ \begin{array}{ccc} k_1 & k_2 & K\\
                          j_4 & j_1 & j_2 \end{array} \right\}
\left\{ \begin{array}{ccc} k'_1 & k'_2 & K'\\
                          j'_4 & j'_1 & j'_2 \end{array} \right\}
\left\{ \begin{array}{ccc} k_1 & k'_1 & k\\
                           k_2 & k'_2 & k'\\
                           K & K' & k'' \end{array} \right\}\nn\\
&& \times
\Bigl( T^K(j_1j_4) \otimes T^{K'}(j'_1j'_4)\Bigr)^{k''}_{q''}
 |j_a\,m_a\rangle  \otimes |j'_a\,m'_a\rangle \,.
\eea
Using the abbreviation
\be
J=j_1+j_2+j_3+j_4\,, \quad
J'=j'_1+j'_2+j'_3+j'_4\,,
\ee
the expression for the commutator of two coupled SO(3) tensor operators can be
written as follows~:
%
%5
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\bea \lefteqn{
\Bigl[ \Bigl( T^{k_1}(j_1j_2) \otimes T^{k'_1}(j'_1j'_2)\Bigr)^k_q\,,\,
\Bigl( T^{k_2}(j_3j_4) \otimes T^{k'_2}(j'_3j'_4)\Bigr)^{k'}_{q'}\Bigr]= }\nn\\
&& \sum_{K,K',k'',q''}
  (-1)^{K+K'+k''+q''+J+J'}
  [k_1\,k_2\,K\,k'_1\,k'_2\,K'\,k\,k'\,k'']^{1/2}
{\small
\addtolength{\arraycolsep}{-.5\arraycolsep}
\left( \begin{array}{ccc} k & k' & k''\\
                          -q & -q' & q'' \end{array} \right)
\left\{ \begin{array}{ccc} k_1 & k'_1 & k\\
                           k_2 & k'_2 & k'\\
                           K & K' & k'' \end{array} \right\} } \nn\\
&& \times \Biggl( (-1)^{2j_1+2j'_1}\delta_{j_2j_3}\delta_{j'_2j'_3}
\left\{ \begin{array}{ccc} k_1 & k_2 & K\\
                          j_4 & j_1 & j_2 \end{array} \right\}
\left\{ \begin{array}{ccc} k'_1 & k'_2 & K'\\
                          j'_4 & j'_1 & j'_2 \end{array} \right\}
\Bigl( T^K(j_1j_4)\otimes T^{K'}(j'_1j'_4)\Bigr)^{k''}_{q''} \nn\\
&& -(-1)^{k_1+k'_1+k_2+k'_2+K+K'}(-1)^{2j_3+2j'_3}\delta_{j_1j_4}\delta_{j'_1j'_4}
\left\{ \begin{array}{ccc} k_1 & k_2 & K\\
                          j_3 & j_2 & j_1 \end{array} \right\}
\left\{ \begin{array}{ccc} k'_1 & k'_2 & K'\\
                          j'_3 & j'_2 & j'_1 \end{array} \right\}\nn\\
&& \times \Bigl( T^K(j_3j_2)\otimes T^{K'}(j'_3j'_2)\Bigr)^{k''}_{q''}\Biggr)\,.
\eea
A case which is of special interest is that for which $j_1=j_2=j_3=j_4=j$ and
$j'_1=j'_2=j'_3=j'_4=j'$. Then, the $j$- or $j'$-dependence of the tensor
operators can be suppressed in the notation. The commutation rule becomes~:
\bea
&& \Bigl[ \Bigl( T^{k_1}\otimes T^{k'_1} \Bigr)^k_q\,,\,
          \Bigl( T^{k_2}\otimes T^{k'_2} \Bigr)^{k'}_{q'} \Bigr] = \nn\\
&& \sum_{K,K',k'',q''}
  (-1)^{2j+2j'+K+K'+k''+q''}
  [k_1\,k_2\,K\,k'_1\,k'_2\,K'\,k\,k'\,k'']^{1/2}
{\small
\addtolength{\arraycolsep}{-.5\arraycolsep}
\left( \begin{array}{ccc} k & k' & k''\\
                          -q & -q' & q'' \end{array} \right)
\left\{ \begin{array}{ccc} k_1 & k'_1 & k\\
                           k_2 & k'_2 & k'\\
                           K & K' & k'' \end{array} \right\} } \nn\\
&&\times
\left\{ \begin{array}{ccc} k_1 & k_2 & K\\
                          j & j & j \end{array} \right\}
\left\{ \begin{array}{ccc} k'_1 & k'_2 & K'\\
                          j' & j' & j' \end{array} \right\}
\Bigl[ 1-(-1)^{k_1+k'_1+k_2+k'_2+K+K'}\Bigr]
\Bigl( T^K \otimes T^{K'}\Bigr)^{k''}_{q''}\,.
\eea
It can be verified that for $k'_1=k'_2=0$, (11) reduces to (2). Equation (11)
can also be seen as the coupled version of formula (2.2) of Reference~7.

From (12) it follows that the set of operators
\bea
\Bigl( T^{k_1} \otimes T^{k'_1}\Bigr)^k_q
& , & k_1=0,1,\ldots,2j,\quad k'_1=0,1,\ldots,2j'\,,\nn\\
&& k=|k_1-k'_1|,\ldots,k_1+k'_1\,,\quad q=-k,\ldots,+k
\eea
close under commutation. In fact, they are the generators of $U(N)$,
%
%6
%
where $N={{(2j+1)}\times(2j'+1)}$. The operator $(T^0\otimes T^0)^0_0$ commutes with all
other operators, and deleting it from the set (13) leaves the generators of
$SU(N)$. The operators $( T^{k_1}\otimes T^0 )^{k_1}_{q_1}$ and
$(T^0\otimes T^{k'_1})^{k'_1}_{q'_1}$ clearly commute, and are the generators
of $U(2j+1)$ and $U(2j'+1)$ respectively. Thus (12) describes the chain
\be
U\Bigl( (2j+1)(2j'+1)\Bigr) \supset U(2j+1) \times U(2j'+1)
\supset SO(3) \times SO(3) \supset SO(3).
\ee
It can also be seen from (12) that the subset of operators in (13) with
$k_1+k'_1$ odd also closes under commutation. They form the generators of
$SO(N)$ if $j$ and $j'$ are both integers or both half-integers, and of
$Sp(N)$ otherwise.

We believe that (11) or (12) is the first expression yielding a $9j$ symbol in
the structure constants of a Lie algebra. Trying to extend the ideas of Refs.
6, 7 and 8, where the identification of exceptional Lie algebras lead to an
explanation of non-trivial zeros of $6j$ symbols appearing in (2), one would at
first sight expect to be in a position to relate non-trivial zeros of $9j$
symbols to exceptional Lie algebras realised as subalgebras of
(11) or (12). However, this happens not to be the case since the internal
$j$-values ($j$ and $j'$) do not appear in the $9j$ symbol. An example can
clarify the situation. From Ref.7 one can deduce that the generators of the
exceptional Lie algebra $F_4$ can be written as follows~:
\bea
L^1_q &=& \bigl( T^1(1,1)\otimes T^0(3,3)\bigr)^1_q
            + \sqrt{5\over7}\bigl( T^1(2,2)\otimes T^0(0,0)\bigr)^1_q\,,\nn\\[.3cm]
K^1_q &=& \bigl( T^0(1,1)\otimes T^1(3,3) \bigr)^1_q\,,\nn\\[.3cm]
P^5_q &=& \bigl( T^0(1,1) \otimes T^5(3,3)\bigr)^5_q\,,\nn\\[.3cm]
G^k_q &=& \bigl( T^2(1,1)\otimes T^3(3,3)\bigr)^k_q
           + {1\over \sqrt{2}}\Bigl( \bigl( T^2(1,2)\otimes T^3(3,0)\bigr)^k_q
           + \bigl( T^2(2,1)\otimes T^3(0,3)\bigr)^k_q\Bigr)\,,\nn\\
      &&   \qquad\qquad\qquad\qquad\qquad\qquad\qquad\qquad\qquad\qquad
           k=1,2,3,4,5.
\eea
%
%7
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Using (11), one can calculate the commutator $\Bigl[ G^k_q\,,\,G^{k'}_{q'}\Bigl]$,
and require that it should close as a linear combination of the operators (15).
In particular, $\Bigl[ G^1_q\,,\,G^3_q\Bigr]$ yields a term in
$\bigl( T^2(1,1) \otimes T^5(3,3)\bigr)^3_{q''}$\,, with a factor proportional to
\be
\left\{ \begin{array}{ccc}
   2 & 3 & 1\\
   2 & 3 & 3\\
   2 & 5 & 3 \end{array} \right\}
\left( \left\{ \begin{array}{ccc}
   2 & 2 & 2\\
   1 & 1 & 1 \end{array} \right\}
   \left\{ \begin{array}{ccc}
   3 & 3 & 5\\
   3 & 3 & 3 \end{array} \right\} +
   {1\over 2} \left\{ \begin{array}{ccc}
   2 & 2 & 2\\
   1 & 1 & 2 \end{array} \right\}
   \left\{ \begin{array}{ccc}
   3 & 3 & 5\\
   3 & 3 & 0 \end{array} \right\} \right)\,.
\ee
As this term should vanish, (16) must be zero. At first sight one would deduce
\be
\left\{ \begin{array}{ccc}
   2 & 3 & 1\\
   2 & 3 & 3\\
   2 & 5 & 3 \end{array} \right\} =0 \,,
\ee
which is indeed valid. However, it so happens that the expression in terms of
$6j$ symbols is also equal to zero\,! Since this expression appears also in other
terms, it would be wrong to deduce the vanishing of a $9j$ symbol from (16).
In fact, all one can say from (16) and similar expressions is that the relation
\be
\left\{ \begin{array}{ccc}
   2 & 2 & K\\
   1 & 1 & 1 \end{array} \right\}
   \left\{ \begin{array}{ccc}
   3 & 3 & K'\\
   3 & 3 & 3 \end{array} \right\} +
   {1\over 2}
   \left\{ \begin{array}{ccc}
   2 & 2 & K\\
   1 & 1 & 2 \end{array} \right\}
   \left\{ \begin{array}{ccc}
   3 & 3 & K'\\
   3 & 3 & 0 \end{array} \right\}=0
\ee
holds for $(K,K')\in \{(0,3),(1,2),(1,4),(1,6),(2,1),(2,5)\}$ (and one can also
deduce the vanishing of the $6j$ symbol discussed in Ref.7). So the closure of
the exceptional Lie algebra involves the vanishing of those symbols (or
combination of symbols) which have internal $j$-values as arguments.
\vskip 5mm
\noindent {\bf Acknowledgements}

One of us (K.S.R.) wishes to thank the N.F.W.O.\ Belgium for financial
support and the Rijksuniversiteit Gent for its hospitality. The authors
thank Dr.\ H.~De Meyer for helpful discussions.

\newpage
%
%8
%
\noindent {\bf References}.
\begin{enumerate}
\item J.P. Elliott, {\em Proc. R. Soc. London, Ser. A} {\bf 245}, 128 (1958).
\item B.R. Judd, {\em Operator Techniques in Atomic Spectroscopy} (McGraw-Hill,
New York, 1963), p. 102.
\item G. Vanden Berghe and H. De Meyer, {\em J. Math. Phys.} {\bf 25}, 772 (1984).
\item M. Rotenberg, R. Bivins, N. Metropolis and J.K. Wooten Jr., {\em The $3j$
  and $6j$ symbols} (The Technology Press, MIT, Cambridge, 1959).
\item L.C. Biedenharn and J.D. Louck, The Racah-Wigner Algebra in Quantum Theory,
{\em Encyclopedia of Mathematics} {\bf 9} (Addison-Wesley, Massachusetts, 1981),
chapter~5, Topic 10.
\item J. Van der Jeugt, G. Vanden Berghe and H. De Meyer, {\em J. Phys. A~:
Math. Gen.} {\bf 16}, 1377 (1983).
\item H. De Meyer, G. Vanden Berghe and J. Van der Jeugt, {\em J. Math. Phys.}
{\bf 25}, 751 (1984).
\item G. Vanden Berghe, H. De Meyer and J. Van der Jeugt, {\em J. Math. Phys.}
{\bf 25}, 2585 (1984).
\item A. de--Shalit and I. Talmi, {\em Nuclear Shell Theory} (Academic Press,
N.Y., 1963), p.~517.
\end{enumerate}















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